The Behaviors of Ferro-Magnetic Nano-Particles In and Around Blood Vessels under Applied Magnetic Fields

摘要

In magnetic drug delivery, therapeutic magnetizable particles are typically injected into the blood stream and magnets are then used to concentrate them to disease locations. The behavior of such particles in-vivo is complex and is governed by blood convection, diffusion (in blood and in tissue), extravasation, and the applied magnetic fields. Using physical first-principles and a sophisticated vessel-membrane-tissue (VMT) numerical solver, we comprehensively analyze in detail the behavior of magnetic particles in blood vessels and surrounding tissue. For any blood vessel (of any size, depth, and blood velocity) and tissue properties, particle size and applied magnetic fields, we consider a Krogh tissue cylinder geometry and solve for the resulting spatial distribution of particles. We find that there are three prototypical behaviors (blood velocity dominated, magnetic force dominated, and boundary-layer formation) and that the type of behavior observed is uniquely determined by three non-dimensional numbers (the magnetic-Richardson number, mass Péclet number, and Renkin reduced diffusion coefficient). Plots and equations are provided to easily read out which behavior is found under which circumstances (, , , and ). We compare our results to previously published in-vitro and in-vivo magnetic drug delivery experiments. Not only do we find excellent agreement between our predictions and prior experimental observations, but we are also able to qualitatively and quantitatively explain behavior that was previously not understood.Magnetic nano-particle behaviors as a function of the magnetic-Richardson and Renkin reduced diffusion coefficient non-dimensional numbers. The mass Péclet number was held constant. Three regions are shown: the magnetic dominated region at the top (the thin solid blue region); the velocity (Stokes drag) dominated region on the left (dashed lines region); and the boundary layer formation region on the right (wavy lines region). Notice that boundary layer build-up behavior is still possible even if the magnetic force is just 0.01% of the Stokes drag force at the vessel centerline, i.e. at >Ψ ≥ 0.0001. The boundary between the velocity and boundary layer build-up regions is diffuse as shown schematically by the thickness of the fuzzy gray line separating them.